Monte Carlo Simulation of Coil-to-Globule Transition of Compact Polymer Chains: Role of Monomer Interacting

Coil-to-globule transitions are fundamental problems existing in polymer science for several decades; however, some features are still unclear, such as the effect of chain monomer interaction. Herein, we use Monte Carlo simulation to study the coil-to-globule transition of simple compact polymer chains. We first consider the finite-size effects for a given monomer interaction, where the short chain exhibits a one-step collapse while long chains demonstrate a two-step collapse, indicated by the specific heat. More interestingly, with the decrease of chain monomer interaction, the critical temperatures marked by the peaks of heat capacity shift to low values. A closer examination from the energy, mean-squared radius of gyration and shape factor also suggests the lower temperature of coil-to-globule transition.     

Stoichiometric polyelectrolyte complexes as comb copolymers

The collapse behavior of a single comblike copolymer chain has been studied by Monte Carlo simulations. It has been supposed that the solvent is good for the side chains but the solvent quality for the backbone chain changes. It has been shown that depending on the structural parameters of the comb copolymer (the lengths of the backbone and side chains, grafting density of the side chains) various thermodynamically stable morphologies of the collapsed backbone chain can be realized. In addition to ordinary spherical globule we have observed elongated structures as well as necklace-like conformations. The proposed model can be used to describe conformational behavior of stoichiometric complexes between block copolymers with a polyelectrolyte short block and oppositely charged linear homopolymers.     

Development of knotting during the collapse transition of polymers

A dynamic Monte Carlo simulation of the collapse transition of polymer chains is presented. The chains are represented as self-avoiding walks on the simple cubic lattice with a nearest-neighbor contact potential to model the effect of solvent quality. The knot state of the chains is determined using the knot group procedure presented in the accompanying paper. The equilibrium knot spectrum and the equilibrium rms radius of gyration as functions of the chain length and the contact potential are reported. The collapse transition was studied following quenches from good-to poor-solvent conditions. Our results confirm the prediction that the newly formed globule is not yet at equilibrium, since it has not yet achieved its equilibrium knot spectrum. For our model system, the relaxation of the knot spectrum is about an order of magnitude slower than that of the radius of gyration. The collapse transition is also studied for a model in which both ends of the chain remain in good-solvent conditions. Over the time scale of these simulations, knot formation is frustrated in this inhomogeneous model, verifying that the mechanism of knotting is the tunneling of chain ends in and out of the globule.     

A Monte Carlo study of the coil-to-globule transition of model polymer chains near an attractive surface

When a polymer chain in solution interacts with an atomically smooth solid substrate, its conformational properties are strongly modified and deviate substantially from those of chains in bulk. In this work, the interplay of two competing transitions that affect the conformations of polymer chains near an energetically attractive surface is studied by means of Monte Carlo simulations on a cubic lattice. The transition from an extended to a compact conformation of a polymer chain near an attractive wall, as solubility deteriorates, exhibits characteristics akin to the “coil-to-globule” transition in bulk. An effective θ-temperature is determined. Its role as the transition point is confirmed in a variety of ways. The nature of the coil-to-compact transition is not qualitatively different from that in the bulk. Adsorbed polymer chains may assume “globular” or “pancake” configurations depending on the competition among adsorption strength, cohesive energy, and entropy. In a very relevant range of conditions, the dependence of the adsorbate thickness on chain-length is intermediate between that of 3-d (“semidroplets”) and 2-d (“pancake”) objects. The focus of this study is on rather long polymer chains. Several crucial features of the transitions of the adsorbed chains are N-dependent and various aspects of the adsorption and “dissolution” process are manifested clearly only at the “long chain” limit. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2462–2476, 2009     

Simulation of Copolymer Bottle-Brushes

Summary: The structure of bottle-brush polymers with a rigid backbone and flexible side chains is studied in three dimensions, varying the grafting density, the side chain length, and the solvent quality. Some preliminary results of theoretical scaling considerations for one-component bottle-brush polymers in a good solvent are compared with Monte Carlo simulations of a simple lattice model. For the simulations a variant of the pruned-enriched Rosenbluth method (PERM) allowing for simultaneous growth of all side chains in the Monte Carlo sampling is employed. For a symmetrical binary (A,B) bottle-brush polymer, where two types (A,B) of flexible side chains are grafted with one chain end to the backbone in an alternating way, varying repulsive binary interactions between unlike monomers and the solvent quality, it is found that phase separation into an A-rich part of the cylindrical molecule and a B-rich part can occur only locally. Long range order (in the direction of the backbone) does not occur, and hence the transition from the randomly mixed state of the bottle-brush to the phase-separated structure is strongly rounded, in contrast to the corresponding mean field predictions of a sharp transition to a “Janus cylinder” phase-separated structure. This lack of a phase transition can be understood from an analogy with spin models in one dimension. By estimating the correlation length for this phase separation along the backbone as a function of side chain length and solvent quality, we present strong evidence that no sharp phase transition occurs.     

Static and dynamic properties of tethered chains at adsorbing surfaces: a Monte Carlo study

We present extensive Monte Carlo simulations of tethered chains of length N on adsorbing surfaces, considering the dilute case in good solvents, and analyze our results using scaling arguments. We focus on the mean number M of chain contacts with the adsorbing wall, on the chain's extension (the radius of gyration) perpendicular and parallel to the adsorbing surface, on the probability distribution of the free end and on the density profile for all monomers. At the critical adsorption strength epsilon(c) one has M(c) approximately N(phi), and we find (using the above results) as best candidate phi to equal 0.59. However, slight changes in the estimation of epsilon(c) lead to large deviations in the resulting phi; this might be a possible reason for the difference in the phi values reported in the literature. We also investigate the dynamical scaling behavior at epsilon(c), by focusing on the end-to-end correlation function and on the correlation function of monomers adsorbed at the wall. We find that at epsilon(c) the dynamic scaling exponent a (which describes the relaxation time of the chain as a function of N) is the same as that of free chains. Furthermore, we find that for tethered chains the modes perpendicular to the surface relax quicker than those parallel to it, which may be seen as a splitting in the relaxation spectrum.     

Computer Simulation of Adsorbed Polymer Chains with a Different Molecular Architecture

Simulations of simple models of polymer chains were carried out by the means of the dynamic Monte Carlo method. The model chains were confined to a simple cubic lattice. Three different chain architectures were studied: linear, star‐branched and ring chains. The polymer model chain interacted with an impenetrable surface with a simple contact attractive potential. It was found that size parameters of all these polymers obey scaling laws. The temperatures of the transitions from weakly to strongly adsorbed chain were determined. It was shown for weakly adsorbed chains that ring polymers are always ca. 50% more adsorbed than linear and star‐branched ones. The properties of adsorbed linear and star‐branched polymers are very similar in the length of chain and the strength of adsorption studied. Strongly adsorbed ring polymers are still more adsorbed but differences between all kinds of chains become less pronounced.     

磁性高分子链的磁性质和构象性质的Monte Carlo模拟

采用退火 (Annealing)MonteCarlo方法 ,从高温到低温顺序模拟了简立方格点上考虑最近邻Ising相互作用的磁性高分子链在不同温度的磁性质和构象性质 .磁性高分子链在低温下存在自发磁矩 ,无限长链的临界温度Tc=1 77± 0 0 5J kB.在临界温度附近 ,高分子链经历了从伸展的无规线团到紧缩球体的塌缩相变 .对链的尺寸、形状、近邻数及能量的分析表明 ,高分子链的构象性质从温度Tc=1 77开始发生较明显的变化 ,这表明高分子Ising链的相变是Ising相互作用和链节运动协同作用的结果 .     

Concentration and saturation effects of tethered polymer chains on adsorbing surfaces

We consider end-grafted chains at an adsorbing surface under good solvent conditions using Monte Carlo simulations and scaling arguments. Grafting of chains allows us to fix the surface concentration and to study a wide range of surface concentrations from the undersaturated state of the surface up to the brushlike regime. The average extension of single chains in the direction parallel and perpendicular to the surface is analyzed using scaling arguments for the two-dimensional semidilute surface state according to Bouchaud and Daoud [J. Phys. (Paris) 48, 1991 (1987)]. We find good agreement with the scaling predictions for the scaling in the direction parallel to the surface and for surface concentrations much below the saturation concentration (dense packing of adsorption blobs). Increasing the grafting density we study the saturation effects and the oversaturation of the adsorption layer. In order to account for the effect of excluded volume on the adsorption free energy we introduce a new scaling variable related with the saturation concentration of the adsorption layer (saturation scaling). We show that the decrease of the single chain order parameter (the fraction of adsorbed monomers on the surface) with increasing concentration, being constant in the ideal semidilute surface state, is properly described by saturation scaling only. Furthermore, the simulation results for the chains' extension from higher surface concentrations up to the oversaturated state support the new scaling approach. The oversaturated state can be understood using a geometrical model which assumes a brushlike layer on top of a saturated adsorption layer. We provide evidence that adsorbed polymer layers are very sensitive to saturation effects, which start to influence the semidilute surface scaling even much below the saturation threshold.     

Isotropic-nematic phase transition in athermal solutions of rod-coil diblock copolymers

We present a hybrid method to investigate the isotropic-nematic (I-N) transition in athermal solutions of rod-coil copolymers. This method incorporates the scaled-particle theory for semiflexible chains with two-chain Monte Carlo simulation for the osmotic second virial coefficient and for the angle-dependent excluded volume. We compare the theoretical prediction with Monte Carlo simulations for fused rod-coil copolymers and find good agreement for both the equation of state and the orientational order parameter. The theory is also used to examine the effects of the bond length, the chain length, and the chain composition on orientational ordering in athermal solutions of rod-coil block copolymers. It predicts I-N transition in rod-coil copolymers with fixed rod length but a variable flexible tail in good agreement with experiments.     

Intramolecularly ordered globules and the collapse of short chains

We investigate the poor-solvent collapse of short chains having different stiffness through self-consistent minimization of the intramolecular free energy under the constraint of fixed segment lengths between adjacent beads. At first the chains form the Random Gaussian Globule, where the beads are distributed at random at the same average distance from the centre of mass, while the segments show very little correlation. At a larger attractive potential, this collapsed globule undergoes a transition to one or more ordered compact states, depending on the chain stiffness. Under very strong contraction, all chains are described as a Compact Ordered Globule: the beads are again at a constant average distance from the centre of mass, while the segments jump back and forth at the globule's wall with a very large correlation. At intermediate contraction, the thinner and stiffer chains form the Oscillating Ordered Globule wherein the beads are alternatively distributed on two concentric on two concentric shells. In this case, we also find metastable states with nonsymmetrical conformations of the chain with respect to its ends. We also briefly discuss the thermodynamics of the coil-globule and globule-globule transitions, showing that in long polymer chains these ordered conformations cannot involve the whole chain. However, we suggest that they might still be found as local globules that form for kinetic reasons.     

Semiflexible polymers grafted to a solid planar substrate: changing the structure from polymer brush to "polymer bristle"

Monte Carlo simulations are presented for a coarse-grained model of polymer brushes with polymers having a varying degree of stiffness. Both linear chains and ring polymers grafted to a flat structureless non-adsorbing substrate surface are considered. Applying good solvent conditions, it is shown that with growing polymer stiffness the brush height increases significantly. The monomer density profiles for the case of ring polymers (chain length N(R) = 64) are very similar to the case of corresponding linear chains (N(L) = 32, grafting density larger by a factor of two) in the case of flexible polymers, while slight differences appear with increasing stiffness. Evidence is obtained that the chain dynamics in brushes is slowed down dramatically with increasing stiffness. Very short stiff rings (N(R) ≤ 16) behave like disks, grafted to the substrate such that the vector, perpendicular to the disk plane, is oriented parallel to the substrate surface. It is suggested that such systems can undergo phase transitions to states with liquid crystalline order.     

Solvent response of mixed polymer brushes

We have performed classical density functional theory calculations to study the behavior of mixed polymer brushes tethered to a planar surface. We assume no lateral segregation of the polymer at the grafting density studied and consider an implicit solvent. For a binary mixture of short and long athermal polymer chains, the short chain is compressed while the long chain is stretched compared with corresponding pure polymer chains at the same grafting density, which is consistent with simulation. This results from configurational entropy effects. Furthermore, we add a mean-field interaction for each polymer brush to simulate their different response towards a solvent. The long chain is forced to dislike the solvent more than the short chain. Through the interplay between the solvent effects and configurational entropy effects, a switch of the polymer brush surface (or outer) layer is found with increasing chain length of the long chain. The transition chain length (long chain) increases with increasing the solvent selectivity, and decreases with increasing the grafting density of the long chain. These results can provide guidance for the design of smart materials based on mixed polymer brushes.     

Properties of Star-Branched Polymer Chains Near a Surface

We considered two model systems of star-branched polymers near an impenetrable surface. The model chains were constructed on a simple cubic lattice. Each star polymer consisted of f = 3 arms of equal length and the total number of segments was up to 799. The excluded volume effect was included into these models only and therefore the system was studied at good solvent conditions. In the first model system polymer chain was terminally attached with one arm to the surface. The grafted arm could slide along the surface. In the second system the star-branched chain was adsorbed on the surface and the strength of adsorption was were varied. The simulations were performed using the dynamic Monte Carlo method with local changes of chain conformations. The internal and local structures of a polymer layer were determined. The lateral diffusion and internal mobility of star-branched chains were studied as a function of strength of adsorption and the chain length. The lateral diffusion and internal mobility of star-branched chains were studied as a function of strength of adsorption and the chain length. It was shown that the behavior of grafted and weakly adsorbed chains was similar to that of a free three-dimensional polymer, while the strongly adsorbed chains behave as a two-dimensional system.     

Studies on the behavior of nanoconfined homopolymers with cyclic chain architecture

We have performed Monte Carlo simulations to study the effect of cyclic architecture on the behavior of homopolymer chains under several conditions of confinement. The collapse of the rings in two stages, a coil-to-globule and a liquidlike-to-solidlike transition, was observed even at extreme confinement. Both transitions were observed at lower temperatures than for linear chains of the same length, 2%-5% lower for unconfined systems, and 10%-15% lower for wall separations below three bond lengths due to the effect of confinement. When the plates separation approached the two-dimensional regime, the coil-to-globule transition shifted to lower temperatures. The inverse trend was observed when the chain length was increased. In the collapsed state, the average size and conformations of linear and cyclic molecules of same length were similar independently of confinement. At temperatures near the coil-to-globule transition, the radius of gyration of unconfined linear chains, [R(g)(2)](linear), became larger than for the cyclic chains, [R(g)(2)](cyclic), and this difference increased considerably with confinement. The radius of gyration ratio [R(g)(2)](linear)/[R(g)(2)](cyclic) in this region decreased rapidly. The decrease was more pronounced and occurred at lower temperatures for slit width confinements. At higher temperatures, in the coil state, the radius of gyration ratio became nearly constant for a given separation, and varied from 0.56 for unconfined systems to 0.47 when the chain was completely confined between the walls. This reduction was attributed to the higher increase in the average size of linear chains with confinement when compared with cyclic chains, due to architectural restrictions.     

A quantitative study of tethered chains in various solution conditions using Langmuir diblock copolymer monolayers

This article summarizes our investigations of tethered chain systems using Langmuir monolayers of poly(dimethylsiloxane)‐polystyrene (PDMS‐PS) diblock copolymers on organic liquids. In this system, the PDMS block adsorbs strongly to the air surface while the PS block dangles into the subphase liquid. The air surface can be made either repulsive or attractive for the tethered PS chain segments by choosing a subphase liquid which has a surface tension less than or greater than that of PS, respectively. The segment profile of the PS block is determined by neutron reflection as a function of the surface density, the molecular weights of the PS and PDMS blocks, and the solution conditions. We cover the range of reduced surface density (Σ ) characteristic of the large body of data in the literature for systems of chains tethered onto solid surfaces from dilute solution in good or theta solvent conditions (Σ < 12). We emphasize quantitative comparisons with analytical profile forms and scaling predictions. We find that the strong‐stretching limit assumed in analytical self‐consistent field calculations (SCF) and scaling theories is not valid over this Σ range. On the other hand, over a large portion of this range (Σ ⪇ 5) tethered chain profiles are well described by a renormalization group theory for weakly interacting or noninteracting chains. Simultaneous with the study of the profile form, the free energy of the tethered chains is examined through the surface tension. A strong increase in the surface pressure is observed with increasing surface density which determines the maximum surface density which can be achieved. This effect is attributed to a combination of higher order osmotic interactions and configurational constraints. This effect may explain several outstanding discrepancies regarding the adsorption of end‐functionalized chains and diblock copolymers onto solid surfaces.     

Conformation studies on sol-gel transition in triblock copolymer solutions

The gelation of physically associating triblock copolymers in a good solvent was investigated by means of the Monte Carlo simulation and a gelation process based on the conformation transition of the copolymer that was described in detail. In our simulative system, it has been found that the gelation is closely related with chain conformations, and there exist four types of chains defined as free, dangling, loop, and bridge conformations. The copolymer chains with different conformations contribute to the formation of gel in different ways. We proposed a conformational transition model, by which we evaluated the role of these four types of chains in sol-gel transition. It was concluded that the free chains keeping the conformation transition equilibrium and the dangling conformation being the hinge of conformation transition, while the chain with loop conformation enlarges the size of the congeries and the chain with bridge conformations binds the congeries consisted of the copolymer chains. In addition, the effects of temperature and concentration on the physical gelation, the association of the copolymer congeries, and the copolymer chain conformations' distribution were discussed. Furthermore, we employed the structure factor analysis to study the association of copolymer conformations and long-range order of the simulation system and found our results are in agreement with the previous experimental conclusions.     

The phase behavior of polyethylene ring chains

The equilibrium properties of an isolated polyethylene ring chain are studied by using molecular dynamics (MD) simulations. The results of an 80-bond linear chain are also presented, which are in agreement with previous studies of square-well chains and Lennard-Jones (LJ) homopolymers. Mainly, we focus on the collapse of polyethylene ring chains. At high temperatures, a fully oblate structure is observed for the ring chains with different chain lengths. For such an oblate structure, a shape factor of delta(*)=0.25 and a rodlike scaling relation between the radius of gyration and chain lengths could be deduced easily in theory, and the same results are obtained by our MD simulations. Such an oblate structure can be obtained by Monte Carlo simulation only for sufficient stiff ring chains. When the temperature decreases, an internal energy barrier is observed. This induces a strong peak in the heat capacity, denoting a gas-liquid-like transition. This energy barrier comes mainly from the local monomer-monomer interactions, i.e., the bond-stretching, the bond-bending, and the torsion potentials. A low temperature peak is also observed in the same heat capacity curve, representing a liquid-solid-like transition. These numerical simulation results support a two-stage collapse of polyethylene ring chains; however, the nature should be different from the square-well and LJ ring chains.     

Conformational studies of bottle-brush polymers absorbed on a flat solid surface

The adsorption of a bottle-brush polymer end-grafted with one chain end of its backbone to a flat substrate surface is studied by Monte Carlo simulation of a coarse-grained model, that previously has been characterized in the bulk, assuming a dilute solution under good solvent conditions. Applying the bond fluctuation model on the simple cubic lattice, we vary the backbone chain length N(b) from N(b)=67 to N(b)=259 effective monomeric units, the side chain length N from N=6 to N=48, and set the grafting density to σ=1, i.e., parameters that correspond well to the experimentally accessible range. When the adsorption energy strength ? is varied, we find that the adsorption transition (which becomes well-defined in the limit N(b)→∞, for arbitrary finite N) roughly occurs at the same value ?(c) as for ordinary linear chains (N=0), at least within our statistical errors. Mean square end-to-end distances and gyration radii of the side chains are obtained, as well as the monomer density profile in the direction perpendicular to the adsorbing surface. We show that for longer side chains the adsorption of bottle-brushes is a two-step process, the decrease of the perpendicular linear dimension of side chains with adsorption energy strength can even be nonmonotonic. Also, the behavior of the static structure factor S(q) is analyzed, evidence for a quasi-two-dimensional scaling is presented, and consequences for the interpretation of experiments are discussed.     

Conformational behavior of polymers adsorbed on nanotubes

The importance of hydrophobic interactions in determining polymer adsorption and wrapping of carbon nanotubes is still under debate. In this work, we concentrate on the effect of short-ranged weakly attractive hydrophobic interactions between polymers and nanotubes (modeled as an infinitely long and smooth cylindrical surface), neglecting all other interactions apart for chain flexibility. Using coarse-grained Monte Carlo simulation of such simplified systems, we find that uniform adsorption and wrapping of the nanotube occur for all degrees of chain flexibility for tubes with sufficiently large outer radii. However, the adsorbed conformations depend on chain stiffness, ranging from randomly adsorbed conformations of the flexible chain to perfect helical or multihelical conformations (in the case of more concentrated solutions) of the rigid chains. Adsorption appears to occur in a sequential manner, wrapping the nanotube nearly one monomer at a time from the point of contact. Once adsorbed, the chains travel on the surface of the cylinder, retaining their helical conformations for the semiflexible and rigid chains. Our findings may provide additional insight to experimentally observed ordered polymer wrapping of carbon nanotubes.